Method and system for utilizing space-time and space-frequency codes for multi-input multi-output frequency selective fading channels

ABSTRACT

A communication system for transmitting encoded signals over a communication channel is disclosed. The system includes a transmitter, which has a source that outputs a message signal. The transmitter also includes an encoder that generates a code word in response to the message signal. The code word has a construction that defines a plurality of paths associated with an intersymbol interference (ISI) environment of the communication channel, wherein the code word achieves a diversity based upon the number of transmit antennas and the number of ISI paths. Further, the transmitter includes a modulator that modulates the code word for transmission over the communication channel, and multiple antennas that transmit the modulated code word over the communication channel. The system encompasses a receiver that receives the transmitted code word via a number of receive antennas.

CROSS-REFERENCES TO RELATED APPLICATION

This application is a continuation of U.S. patent application bearingSer. No. 11/231,691, filed Sep. 21, 2005, now U.S. Pat. No. 7,315,570entitled “Method and System for Utilizing Space-Time and Space-FrequencyCodes for Multi-Input Multi-Output Frequency Selective Fading Channels”,inventors: Hesham El-Gamal and Roger Hammons, which is a divisional ofU.S. patent application bearing Ser. No. 10/012,056, filed Nov. 5, 2001,now U.S. Pat. No. 7,010,053 entitled “Method and System for UtilizingSpace-Time and Space-Frequency Codes for Multi-Input Multi-OutputFrequency Selective Fading Channels”, inventors: Hesham El-Gamal andRoger Hammons, which claims priority from U.S. provisional applicationbearing Ser. No. 60/246,024, filed Nov. 6, 2000; the entire contents ofall applications are incorporated herein by this reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to coding in a communication system, andis more particularly related to space-time codes that exploit multipleforms of diversity.

2. Discussion of the Background

Given the constant demand for higher system capacity of wirelesssystems, multiple antenna systems have emerged to increase systembandwidth vis-à-vis single antenna systems. In multiple antenna systems,data is parsed into multiple streams, which are simultaneouslytransmitted over a corresponding quantity of transmit antennas. At thereceiving end, multiple receive antennas are used to reconstruct theoriginal data stream. To combat the detrimental effects of thecommunication channel, communication engineers are tasked to developchannel codes that optimize system reliability and throughput in amultiple antenna system.

To minimize the effects of the communication channel, which typically isRayleigh, space-time codes have been garnered significant attention.Rayleigh fading channels introduce noise and attenuation to such anextent that a receiver may not reliably reproduce the transmitted signalwithout some form of diversity; diversity provides a replica of thetransmitted signal. Space-time codes are two dimensional channel codesthat exploit spatial transmit diversity, whereby the receiver canreliably detect the transmitted signal. Conventional designs ofspace-time codes have focused on maximizing spatial diversity inquasi-static fading channels and fast fading channels. However, realcommunication systems exhibit channel characteristics that are somewherebetween quasi-static and fast fading. Accordingly, such conventionalspace-time codes are not optimized.

Further, other approaches to space-time code design assume that channelstate information (CSI) are available at both the transmitter andreceiver. Thus, a drawback of such approaches is that the designrequires the transmitter and receiver to have knowledge of the CSI,which increases implementation costs because of the need for additionalhardware. Moreover, these approaches view the transmit diversityattending the use of space-time codes as a substitute for timediversity; consequently, such space-time codes are not designed to takeadvantage of other forms of diversity.

Notably, information theoretic studies have shown that spatial diversityprovided by multiple transmit and/or receive antennas allows for asignificant increase in the capacity of wireless communication systemsoperated in a flat Rayleigh fading environment [1] [2]. Following thisobservation, various approaches for exploiting this spatial diversityhave been proposed. In one approach, channel coding is performed acrossthe spatial dimension as well

SUMMARY OF THE INVENTION

The present invention addresses the above stated needs by providingspace-time codes that exploit the multipath nature of the communicationchannel, which exhibits characteristics of a multi-input multi-output(MIMO) selective block fading channel. The code have a construction thatdefines a intersymbol interference (ISI) paths in the communicationchannel, wherein the code achieves a diversity based upon the number oftransmit antennas and the number of ISI paths.

According to one aspect of the invention, a method for transmittingencoded signals over a communication channel of a communication systemis provided. The method includes receiving a message signal.Additionally, the method includes generating a code word in response tothe message signal for transmission over the communication channel via aplurality of transmit antennas. The code word has a construction thatdefines a plurality of paths associated with an intersymbol interference(ISI) environment of the communication channel, wherein the code wordachieves a diversity that is based upon the number of transmit antennasand the number of ISI paths. Under this approach, spatial diversity andtemporal diversity are enhanced, without sacrificing transmission rate.

According to another aspect of the invention, an apparatus for encodingsignals for transmission over a communication channel of a communicationsystem is provided. The apparatus includes a source that is configuredto output a message signal. The apparatus also includes an encoder thatis configured to generate code word in response to the message signalfor transmission over the communication channel via a plurality oftransmit antennas. The code word has a construction that defines aplurality of paths associated with an intersymbol interference (ISI)environment of the communication channel. The code word achieves adiversity that is based upon the number of transmit antennas and thenumber of ISI paths. The above arrangement advantageously improvessystem throughput and system reliability of a communication system.

According to one aspect of the invention, an apparatus for encodingsignals for transmission over a communication channel of a communicationsystem is provided. The apparatus includes means for receiving a messagesignal. Additionally, the apparatus includes means for generating a codeword in response to the message signal for transmission over thecommunication channel via a plurality of transmit antennas. The codeword has a construction that defines a plurality of paths associatedwith an intersymbol interference (ISI) environment of the communicationchannel, wherein the code word achieves a diversity that is based uponthe number of transmit antennas and the number of ISI paths. The abovearrangement advantageously provides increased system capacity.

According to another aspect of the invention, a communication system fortransmitting encoded signals over a communication channel is disclosed.The system includes a transmitter, which has a source that is configuredto output a message signal. The transmitter also includes an encoderthat is configured to generate a code word in response to the messagesignal. Further, the transmitter includes a modulator that is configuredto modulate the code word for transmission over the communicationchannel, and a plurality of transmit antennas that are configured totransmit the modulated code word over the communication channel. Thecode word has a construction that defines a plurality of pathsassociated with an intersymbol interference (ISI) environment of thecommunication channel, wherein the code word achieves a diversity basedupon the number of transmit antennas and the number of ISI paths. Thesystem encompasses a receiver that includes a plurality of receiveantennas, in which the receiver is configured to receive the transmittedcode word via a plurality of receive antennas. The above arrangementadvantageously maximizes spatial and temporal diversity.

According to another aspect of the invention, a waveform signal fortransmission over a communication channel of a communication system isdisclosed. The waveform signal includes a code word that is based upon amessage signal. The code word being generated for transmission over thecommunication channel via a plurality of transmit antennas, wherein thecode word has a construction that defines a plurality of pathsassociated with an intersymbol interference (ISI) environment of thecommunication channel. The code word achieves a diversity based upon thenumber of transmit antennas and the number of ISI paths. The aboveapproach minimizes data transmission errors.

In yet another aspect of the invention, a computer-readable mediumcarrying one or more sequences of one or more instructions fortransmitting encoded signals over a communication channel of acommunication system is disclosed. The one or more sequences of one ormore instructions include instructions which, when executed by one ormore processors, cause the one or more processors to perform the step ofreceiving a message signal. Another step includes generating a code wordin response to the message signal for transmission over thecommunication channel via a plurality of transmit antennas. The codeword has a construction that defines a plurality of paths associatedwith an intersymbol interference (ISI) environment of the communicationchannel, wherein the code word achieves a diversity that is based uponthe number of transmit antennas and the number of ISI paths. Thisapproach advantageously maximizes the diversity in the communicationchannel.

In yet another aspect of the present invention, an apparatus forreceiving signals over a communication channel of a communication systemis provided. The apparatus includes a demodulator that is configured todemodulate a signal containing a code word. The code word has aconstruction that defines a plurality of paths associated with anintersymbol interference (ISI) environment of the communication channel.The code word achieves a diversity that is based upon the number oftransmit antennas and the number of ISI paths. The apparatus alsoincludes a decoder that is configured to decode the code word and tooutput a message signal. Under this approach, the effective bandwidth ofthe communication system is increased.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the invention and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1 is a diagram of a communication system configured to utilizespace-time codes, according to an embodiment of the present invention;

FIG. 2 is a diagram of an encoder that generates space-time codes, inaccordance with an embodiment of the present invention;

FIGS. 3A and 3B are diagrams of receivers that employ space-time codesand space-frequency codes, respectively, according to variousembodiments of the present invention;

FIGS. 4A-4G are graphs of simulation results of the space-time codes andspace-frequency codes, according to the embodiments of the presentinvention;

FIG. 5 is a diagram of a wireless communication system that is capableof employing the space-time codes and space-frequency codes, accordingto embodiments of the present invention; and

FIG. 6 is a diagram of a computer system that can perform the processesof encoding and decoding of space-time codes and space-frequency, inaccordance with embodiments of the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following description, for the purpose of explanation, specificdetails are set forth in order to provide a thorough understanding ofthe invention. However, it will be apparent that the invention may bepracticed without these specific details. In some instances, well-knownstructures and devices are depicted in block diagram form in order toavoid unnecessarily obscuring the invention.

Although the present invention is discussed with respect to BinaryPhase-Shift Keying (BPSK) and Quadrature Phase-Shift Keying (QPSK)modulation, the present invention has applicability to other modulationschemes.

FIG. 1 shows a diagram of a communication system configured to utilizespace-time codes, according to an embodiment of the present invention. Adigital communication system 100 includes a transmitter 101 thatgenerates signal waveforms across a communication channel 103 to areceiver 105. In the discrete communication system 100, transmitter 101has a message source that produces a discrete set of possible messages;each of the possible messages have a corresponding signal waveform.These signal waveforms are attenuated, or otherwise altered, bycommunications channel 103. One phenomena of interest is IntersymbolInterference (ISI), in which the channel 103 causes the overlap ofsignal pulses, resulting in the lost of signal orthogonality. Asdescribed with respect to the construction of space-frequency codes, thechannel ISI characteristics are minimized. It is evident that receiver105 must be able to compensate for the attenuation that is introduced bychannel 103.

To assist with this task, transmitter 101 employs coding to introduceredundancies that safeguard against incorrect detection of the receivedsignal waveforms by the receiver 105. To minimize the impact of thecommunication channel 103 on the transmission signals, channel coding isutilized. An algebraic design framework for layered and non-layeredspace-time codes in flat fading channels are in the following: A. R.Hammons Jr. and H. El Gamal. “On the theory of space-time codes for PSKmodulation,” IEEE Trans. Info. Theory, March 2000; and H. El Gamal andA. R. Hammons Jr. “The layered space-time architecture: a newprospective,” IEEE Trans. Info. Theory, 1999; each of which isincorporated herein by reference in its entirety.

Based upon the algebraic design framework for space-time coding in flatfading channels in “On the Theory of Space-Time Codes for PSKModulation,” A. R. Hammons Jr. and H. El Gamal, IEEE Trans. Info.Theory, March 2000, the present invention extends this framework todesign algebraic codes for multi-input multi-output (MIMO) frequencyselective fading channels. The codes, according to the presentinvention, optimally exploit both the spatial and frequency diversityavailable in the channel. Two design approaches with differentcomplexity-versus-diversity advantage trade-offs are considered. Thefirst approach (referred to as “single carrier time domain design”approach or STC (space-time coding)), which is more fully describedbelow in FIG. 3A, uses space-time coding and maximum likelihood (ML)decoding to exploit the multipath nature of the channel. The secondapproach utilizes an orthogonal frequency division multiplexing (OFDM)technique to transform the multi-path channel into a block fadingchannel (referred to as “OFDM based design” approach or SFC(space-frequency coding)); this approach is detailed in the discussionof FIG. 3B. The new algebraic framework, according to one embodiment ofthe present invention, is then used to construct space-frequency codesthat optimally exploit the diversity available in the resulting blockfading channel.

The two approaches, according to the present invention, differ in termsof decoder complexity, maximum achievable diversity advantage, andsimulated frame error rate performance. The first approach requiresrelatively greater complexity at the receiver 105 over the secondapproach, in that the first approach combines algebraic space-timecoding with maximum likelihood decoding to achieve the maximum possiblediversity advantage in MIMO frequency selective channels to achieve thediversity advantage. As a result, this first approach has a relativelylarge trellis complexity, as required by the maximum likelihood receiver105. The second approach utilizes an orthogonal frequency divisionmultiplexing (OFDM) front-end to transform an intersymbol-interference(ISI) fading channel into a flat block fading channel.

FIG. 2 shows a diagram of an encoder that generates space-time codes, inaccordance with an embodiment of the present invention. A transmitter200 is equipped with a channel encoder 203 that accepts input from aninformation source 201 and outputs coded stream of higher redundancysuitable for error correction processing at the receiver 105 (FIG. 1).The information source 201 generates k signals from a discrete alphabet,X′. Encoder 203 generates signals from alphabet Y to a modulator 205.Modulator 205 maps the encoded messages from encoder 203 to signalwaveforms that are transmitted to L_(t) number of antennas 207, whichemit these waveforms over the communication channel 103. Accordingly,the encoded messages are modulated and distributed among the L_(t)antennas 207. The transmissions from each of the L_(t) transmit antennas207 are simultaneous and synchronous.

FIG. 3A shows a diagram of a decoder that decodes space-time codes,according to an embodiment of the present invention. At the receivingside, a receiver 300 includes a demodulator 301 that performsdemodulation of received signals from transmitter 200. These signals arereceived at multiple antennas 303. The signal received at each antenna303 is therefore a superposition of the L_(t) transmitted signalscorrupted by additive white Gaussian noise (AWGN) and the multiplicativeintersymbol interference (ISI) fading. After demodulation, the receivedsignals are forwarded to a decoder 305, which attempts to reconstructthe original source messages by generating messages, X′. Receiver 300,according to one embodiment of the present invention, has a memory 307that stores channel state information (CSI) associated with thecommunication channel 103. Conventional communication systems typicallyrequire that CSI be available at both the transmitter and the receiver.By contrast, the present invention, according to one embodiment, doesnot require CSI at the transmitter 200, thus, providing a more robustdesign.

At the receiver 300, the signal r_(i) ^(j) received by antenna j at timet is given by

$r_{t}^{j} = {{\sqrt{E_{s}}{\sum\limits_{l = 0}^{L_{ISI} - 1}{\sum\limits_{i = 1}^{L_{t}}{\alpha_{l}^{ij}s_{t - 1}^{i}}}}} + n_{t}^{j}}$where √{square root over (E_(s))}, is the energy per transmitted symbol;α_(t) ^(ij) is the complex path gain from transmit antenna i to receiveantenna j for the lth path; L_(ISI) is the length of the channel impulseresponse; s_(t) ^(i) is the symbol transmitted from antenna i at time t;n_(t) ^(j) is the additive white Gaussian noise sample for receiveantenna j at time t. The noise samples are independent samples ofcircularly symmetric zero-mean complex Gaussian random variable withvariance N₀/2 per dimension. The different path gains α_(t) ^(ij) areassumed to be statistically independent.

A space-time code is defined to include an underlying error control codetogether with a spatial parsing formatter. Specifically, an L_(t)×lspace-time code C of size M has an (L_(t)l, M) error control code C anda spatial parser σ that maps each code word vector c ε C to an L_(t)×lmatrix c whose entries are a rearrangement of those of c. The space-timecode C is said to be linear if both C and σ are linear.

It is assumed that the standard parser mapsc =(c ₁ ⁽¹⁾ ,c ₁ ⁽²⁾ , . . . ,c ₁ ^((L) ^(t) ⁾ ,c ₂ ⁽¹⁾ ,c ₂ ⁽²⁾ , . . .,c ₂ ^((L) ^(t) ⁾ , . . . c _(t) ⁽¹⁾ , c _(l) ⁽²⁾ , . . . ,c _(l) ^((L)^(t) ⁾)ε Cto the matrix

$c = \begin{bmatrix}c_{1}^{1} & c_{2}^{1} & \cdots & c_{n}^{1} \\c_{1}^{2} & c_{2}^{2} & \cdots & c_{n}^{2} \\\vdots & \vdots & ⋰ & \vdots \\c_{1}^{L_{t}} & c_{2}^{L_{t}} & \cdots & c_{n}^{L_{t}}\end{bmatrix}$The baseband code word f(c) is obtained by applying the modulationoperator f on the components of c. This modulation operator maps theentries of c into constellation points from the discrete complex-valuedsignaling constellation Ω for transmission across the channel. In thisnotation, it is understood that c_(t) ^((i)) is the code symbol assignedto transmit antenna i at time t and s_(t) ^((i))=f(c_(t) ^((i))).

The diversity advantage of a space-time code is defined as the minimumabsolute value of the slope of any pairwise probability of error versussignal-to-noise ratio curve on a log-log scale. To maximize the spatialdiversity advantage provided by the multiple transmit antenna inquasi-static flat fading MIMO channels, the following rank criterion isutilized [3][4]: for the baseband rank criterion, d=rank(f(c)−f(e)) ismaximized over all pairs of distinct code words c, e ε C. Therefore fullspatial transmit diversity is achieved if and only ifrank(f(c)−f(e))=L_(t) for all pairs of distinct code words c, e ε C. Itshould be noted that in the presence of L_(r) receive antennas 303, thetotal diversity advantage achieved by this code is L_(t)L_(r).

Space-time code constructions for frequency selective fading channels isbased on the concept that in an ISI (intersymbol interference)environment with L_(ISI) paths, a space-time system with L_(t) transmitantennas 207 is equivalent to a space-time system operating in flatfading channel with L_(t)L_(ISI) transmit antenna 207. However, in thisequivalent model the code word matrices are restricted to have a certainspecial structure. This structure is captured in the followingdefinition for the baseband code word matrix in ISI environments:

${f(c)}_{ISI} = \begin{bmatrix}{f(c)} & \underset{\_}{0} & \cdots & \underset{\_}{0} \\\underset{\_}{0} & {f(c)} & \cdots & \underset{\_}{0} \\\vdots & \vdots & ⋰ & \vdots \\\underset{\_}{0} & \underset{\_}{0} & \cdots & {f(c)}\end{bmatrix}$where c is the code word matrix as defined in (2) below, and 0 is theL_(t)×1 all zero vector. From the equivalent model, it is clear that inthe frequency selective fading channels, space-time codes can beconstructed to achieve L_(t)L_(ISI) transmit diversity order. Therefore,the following baseband design criterion for space-time codes in the ISIchannel is established: for ISI baseband rank criterion,d=rank(f_(ISI)(c)−f_(ISI)(e)) is maximized over all pairs of distinctcode words c, e ε C. Full transmit diversity in this scenario is equalto L_(t)L_(ISI), and is achieved if and only ifrank(f_(ISI)(c)−f_(ISI)(e))=L_(t)L_(ISI) for all pairs of distinct codewords c, e ε C.

Next, the binary rank criteria is developed; this criteria facilitatethe construction of algebraic space-time codes for BPSK (BinaryPhase-Shift Keying) and QPSK (Quadrature Phase-Shift Keying) modulatedsystems with an arbitrary number of transmit antennas 207 and channelimpulse response lengths. A new code word matrix c_(ISI) that capturesthe nature of the ISI channel is defined as follows:

$c_{ISI} = \begin{bmatrix}c & \underset{\_}{0} & \cdots & 0 \\0 & c & \cdots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \cdots & c\end{bmatrix}$It is first observed that in generalf(c _(ISI))≠f(c)_(ISI),   (2)sincef(0)≠0However, it is noted the diversity advantage only depends on differencesbetween code words rather than the code words themselves, and thusf(c _(ISI))−f(e _(ISI))=f(c)_(ISI) −f(e)_(ISI)for any signaling constellation. The previous result is the key to thealgebraic space-time constructions developed in this section.

Attention is now turned to the development of BPSK modulated codes,which may be utilized in the communication system 100 of FIG. 1. ForBPSK modulation, elements in c are drawn from the field F={0,1} ofintegers modulo 2. The modulation operator/maps the symbol c_(t) ^((i))ε F to the constellation point s_(t) ^((i))=f(c_(t) ^((i))) ε {−1,1)}according to the rule f(c_(t) ^((i)))=(−1)^(c) ^(t) ^((i)) . The binaryrank criterion for full diversity space-time codes in ISI channels canthus be stated as follows.

With respect to the ISI channel binary rank criterion, it is assumedthat C is a linear L_(t)×l space-time code with underlying binary code Cof length N=L_(t)l operating in an ISI channel with L_(ISI) paths, wherel≧L_(t)L_(ISI). Also, assuming that every non-zero code word ccorresponds to a matrix c_(ISI) of full rank L_(t)L_(ISI) over thebinary field F, then, for BPSK transmission over the frequency selectivequasi-static fading channel 103, the space-time code C achieves fulltransmit diversity L_(t)L_(ISI).

While the previous result was stated for full transmit diversity codes,it readily generalizes to any order of transmit diversity less than orequal to L_(t)L_(ISI). The ISI channel binary rank criterion permits theuse of a stacking construction that establishes an algebraic frameworkfor the design of algebraic space-time codes for MIMO ISI fadingchannels. According to an embodiment of the present invention, the ISIchannel stacking construction, M₁, M₂, . . . , M_(L) _(t) are binarymatrices of dimension k×l,l≧k, and C is the L_(t)×l space-time code ofdimension k including the code word matrices

${c = \begin{bmatrix}{\underset{\_}{x}\; M_{1}} \\{\underset{\_}{x}\; M_{2}} \\\vdots \\{\underset{\_}{x}\; M_{L_{t}}}\end{bmatrix}},$where x denotes an arbitrary k-tuple of information bits and L_(t)<l.The following is denotedM _(n,m) =└O _(L) _(t) _(×(m−1)) M _(n) O _(L) _(t) _(×(L) _(ISI)_(+1−m))┘,where O_(L) _(t) _(×(m−1)) is the L_(t)×(m−1) all zero matrix. Hence, Csatisfies the ISI channel binary rank criterion, and accordingly, forBPSK transmission over the quasi-static fading channel, achieves fulltransmit diversity L_(t)L_(ISI), if and only if M_(1,1), M_(2,1), . . ., M_(L) _(t) _(L) _(ISI) have the property that ∀a₁,a₂, . . . ,a_(L)_(t) ε F: M=a₁M_(1,1) ⊕ a₂M_(2,1) ⊕ . . . ⊕ a_(L) _(t) _(L) _(ISI) M_(L)_(t) _(L) _(ISI) is of full rank k unless a₁= . . . a_(L) _(t) _(L)_(ISI) =0. It is noted that

$c_{ISI} = {\begin{bmatrix}{\underset{\_}{x}\; M_{1,1}} \\{\underset{\_}{x}\; M_{1,2}} \\\vdots \\{\underset{\_}{x}\; M_{L_{t},L_{ISI}}}\end{bmatrix}.}$

The stacking construction is general and applies to block codes as wellas trellis codes. An important example of the stacking construction isgiven by the class of binary convolutional codes. This class isimportant because it allows for a reasonable complexity maximumlikelihood decoder. Let C be the binary, rate l/L_(t), convolutionalcode having transfer function matrix [6]G(D)=└g ₁(D), g ₂(D), . . . ,g_(L) _(t) _(,1)(D), . . . ,g _(L) _(t)_(,L) _(ISI) (D)┘,then the natural space-time code C associated with C is defined toinclude the code word matrices c(D)=G^(T)(D)x(D), where the polynomialx(D) represents the input information bit stream. In other words, forthe natural space-time code, the natural transmission format is adopted,in which the output coded bits generated by g_(i)(D) are transmitted viaantenna i. It is assumed the trellis codes are terminated by tail bits[3]. Thus, if x(D) is restricted to a block of N information bits, thenC is an L_(t)×(N+v) space-time code, where v=max_(1≦i≦L) _(t) {degg_(i)(x)} is the maximal memory order of the convolutional code C. Thefollowing is denotedG _(ISI)(D)=└g _(1,1)(D), g _(2,1)(D), . . . ,g _(L) _(t) _(,1)(D), . .. ,g _(L) _(t) _(,L) _(ISI) (D)┘where g_(n.m)=D^((m−1))g_(n). The following characterizes the result ofthe performance of natural space-time convolutional codes in ISIchannels.

The natural space-time code C associated with the rate 1/L_(t)convolutional code C satisfies the binary rank criterion, and thusachieves full transmit diversity for BPSK transmission in an ISI channelwith L_(ISI) paths, if and only if the transfer function matrixG_(ISI)(D) of C has full rank L_(t)L_(ISI) as a matrix of coefficientsover F. This result stems from the observation that

${\sum\limits_{{1 \leq i \leq L_{t}},{1 \leq j \leq L_{ISI}}}{a_{i,j}{g_{i,j}(D)}{x(D)}}} = 0$for some x(D)≠0 iff

${\sum\limits_{{1 \leq i \leq L_{t}},{1 \leq j \leq L_{ISI}}}{a_{i,j}{g_{i,j}(D)}}} = 0.$This observation readily generalizes to recursive convolutional codes.

The above result extends to convolutional codes with arbitrary rates andarbitrary diversity orders. Since the coefficients of G_(ISI)(D) form abinary matrix of dimension L_(t)L_(ISI)×(v+L_(ISI)), and the column rankmust be equal to the row rank, the result provides a simple bound as tohow complex the convolutional code must be in order to satisfy the fulldiversity ISI channel binary rank criterion.

The maximum diversity order achieved by a space-time code based on anunderlying rate 1/L_(t) convolutional code C with a maximal memory orderv in a L_(ISI) paths ISI channel is v+L_(ISI). This bound shows that,for a fixed trellis complexity, increasing the number of antennas beyond

$L_{t} = \frac{v + L_{ISI}}{L_{ISI}}$will not result in an increase in the diversity advantage. This fact issupported by the results in Table 1, below, which lists the diversityadvantage for BPSK algebraic space-time codes with optimal free distancefor MIMO frequency selective fading channels:

TABLE 1 d for d for d for d for L_(t) ν Connection Polynomials L_(ISI) =1 L_(ISI) = 2 L_(ISI) = 3 L_(ISI) = 4 2 2 5, 7 2 4 5 6 3 64, 74 2 4 6 74 46, 72 2 4 6 8 5 65, 57 2 4 6 8 6 554, 744 2 4 6 8 3 3 54, 64, 74 3 56 7 4 52, 66, 76 3 6 7 8 5 47, 53, 75 3 6 8 9 6 554, 624, 764 3 6 9 10 44 52, 56, 66, 76 4 6 7 8 5 53, 67, 71, 75 4 7 8 9 5 5 75, 71, 73, 65, 575 7 8 9

Because the number of paths is not known a priori at the transmitter200, it is desirable to construct space-time codes that achieve themaximum diversity order for arbitrary number of paths. This leads to thenotion of universal space-time codes that combine the maximum spatialdiversity with the ISI channel frequency diversity whenever available.Within the class of universal space-time codes with maximum diversityadvantage, it is ideal to select the code with the maximum productdistance, which measures the asymptotic coding achieved by the code [3][4].

Although BSPK modulation is discussed, it is recognized that theextension to QPSK modulation can be readily made. The ISI binary rankcriterion and stacking construction for BPSK modulation can begeneralized to obtain similar results for QPSK modulation. As aconsequence of the QPSK ISI binary rank criterion and stackingconstruction, it is observed that the binary connection polynomials ofTable 1 can be used to generate linear, Z₄-valued, rate 1/L_(t)convolutional codes whose natural space-time formatting achieves fullspatial diversity L_(t)L_(ISI) for QPSK modulation. More generally, anyset of Z₄-valued connection polynomials with modulo 2 projections (shownTable 1) may be used. In most cases under consideration, the bestperformance was obtained from the lifted Z₄ codes constructed byreplacing the zero coefficients by twos. This lifting produces the codesin Table 2, which lists Z₄ space-time codes for QPSK modulation in MIMOfrequency selective fading channels.

TABLE 2 L_(t) ν Connection Polynomials 2 1 1 + 2D, 2 + D 2 1 + 2D + D²,1 + D + D² 3 1 + D + 2D² + D³, 1 + D + D² + D³ 4 1 + 2D + 2D² + D³ + D⁴,1 + D + D² + 2D³ + D⁴ 5 1 + D + 2D² + D³ + 2D⁴ + D⁵, 1 + 2D + D² + D³ +D⁴ + D⁵ 3 2 1 + 2D + 2D², 2 + D + 2D², 1 + D + 2D² 3 1 + D + 2D² + D³,1 + D + 2D² + D³, 1 + D + D² + D³ 4 1 + 2D + D² + 2D³ + D⁴, 1 + D +2D² + D³ + D⁴, 1 + D + D² + D³ + D⁴ 5 1 + 2D + 2D² + D³ + D⁴ + D⁵, 1 +2D + D² + 2D³ + D⁴ + D⁵, 1 + D + D² + D³ + 2D⁴ + D⁵ 4 3 1 + 2D + 2D² +2D³, 2 + D + 2D² + 2D³, 2 + 2D + D² + 2D³, 2 + 2D + 2D² + D³ 4 1 + 2D +D² + 2D³ + D⁴, 1 + D + 2D² + D³ + D⁴, 1 + D + 2D² + D³ + D⁴, 1 + D +D² + D³ + D⁴ 5 1 + 2D + D² + 2D³ + D⁴ + D⁵, 1 + D + 2D² + D³ + D⁴ + D⁵,1 + D + D² + 2D³ + 2D⁴ + D⁵, 1 + D + D² + D³ + 2D⁴ + D⁵ 5 4 1 + 2D +2D² + 2D³ + 2D⁴, 2 + D + 2D² + 2D³ + 2D⁴, 2 + 2D + D² + 2D³ + 2D⁴, 2 +2D + 2D² + D³ + 2D⁴, 2 + 2D + 2D² + 2D³ + D⁴ 5 1 + D + D² + D³ + 2D⁴ +D⁵, 1 + D + D² + 2D³ + 2D⁴ + D⁵, 1 + D + D² + 2D³ + D⁴ + D⁵, 1 + D +2D² + D³ + 2D⁴ + D⁵, 1 + 2D + D² + D³ + 2D⁴ + D⁵

The described single carrier time domain design approach requires theuse of a relatively more complex maximum likelihood decoder 305 toaccount for the multi-input multi-output ISI nature of the channel 103.In an exemplary embodiment, this maximum likelihood decoder 305 can berealized using a Viterbi decoder with trellis complexity proportional to2^((L) ^(ISI) ^(+v)) and 4^((L) ^(ISI) ^(+v)) for BPSK and QPSKmodulations, respectively (wherein v is the maximal memory order of theunderlying convolutional code).

If receiver complexity presents an issue, which is conceivable incertain applications, then a second design approach may be implemented.Such an approach uses space-frequency codes. In particular, to reducethe complexity of the receiver 300, an OFDM front-end 313 is utilized totransform the ISI channel into a flat, however, selective fadingchannel. The baseband signal assigned to each antenna 207 is passedthrough an inverse fast Fourier transform (IFFT) before transmission.The transmitted signal from antenna i at the nth interval is given by

${x_{n}^{i} = {\sum\limits_{k = 0}^{N - 1}{s_{k}^{i}{\exp( {{- j}\frac{2\;\pi\;{kn}}{N}} )}}}},$where N is block length. A cyclic prefix of length L_(ISI)−1 is added toeliminate the ISI between consecutive OFDM symbols. At the receiver end,the signal y_(n) ^(j) received by antenna j at time t is given by

$\begin{matrix}{y_{n}^{j} = {{\sqrt{E_{s}}{\sum\limits_{l = 0}^{L_{ISI} - 1}{\sum\limits_{i = 1}^{L_{t}}{\alpha_{l}^{ij}x_{t - 1}^{j}}}}} + n_{t}^{j}}} \\{= {{\sqrt{E_{s}}{\sum\limits_{l = 0}^{L_{ISI} - 1}{\sum\limits_{i = 1}^{L_{t}}{\sum\limits_{k = 0}^{N - 1}{\alpha_{l}^{ij}s_{k}^{j}{\exp( {{- j}\frac{2\pi\;{k( {n - 1} )}}{N}} )}}}}}} + n_{i}^{j}}}\end{matrix}$The fast Fourier transform (FFT) operator is then applied to thereceived signal to yield

$\begin{matrix}{r_{t}^{j} = {\sum\limits_{n = 0}^{N - 1}{y_{k}^{j}{\exp( {{- j}\frac{2\pi\; n\; t}{N}} )}}}} \\{= {{\sum\limits_{i = 1}^{L_{t}}{( {\sum\limits_{l = 0}^{L_{ISI} - 1}{\alpha_{l}^{ij}{\exp( {{- j}\frac{2\pi\; n\; t}{N}} )}}} )s_{t}^{i}}} + N_{t}^{j}}} \\{{= {{\sum\limits_{i = 1}^{L_{t}}{H_{t}^{({ij})}s_{t}^{i}}} + N_{t}^{j}}},}\end{matrix}$where N_(t) ^(j) are independent noise samples of circularly symmetriczero-mean complex Gaussian random variable with variance N₀/2 perdimension. The complex fading coefficients of the equivalent channelmodel H_(t) ^(ij) have the following auto-correlation function:

$\begin{matrix}{{R( {{i_{1} - i_{2}},{j_{1} - j_{2}},{t_{1} - t_{2}}} )} = {E( {H_{t_{1}}^{({i_{1}j_{1}})}H_{t_{2}}^{{({i_{2}j_{2}})}*}} )}} \\{{= {{\delta( {{i_{1} - i_{2}},{j_{1} - j_{2}}} )}{\sum\limits_{l = 0}^{L_{ISI} - 1}{\exp( {{- j}\frac{2\pi\;{l( {t_{1} - t_{2}} )}}{N}} )}}}},}\end{matrix}$where δ(i,j) is the dirac-delta function. It is clear that the fadingcoefficients of the equivalent channel are spatially independent [6] andthat

${R( {0,0,\frac{kN}{L_{ISI}}} )} = 0$for k=1,2, . . . ,L_(ISI)−1. This observation suggests that theequivalent fading channel can be approximated by the piece-wise constantblock fading channel. In this model the code word encompasses L_(ISI)fading blocks. It is assumed that the complex fading gains are constantover one fading block, but are independent from block to block. Anothertype of receiver may be utilized in the event that receiver complexitypresents a key design concern, as shown in FIG. 3B.

FIG. 3B shows a diagram of a receiver that employs space-frequencycodes, according to an embodiment of the present invention. As withreceiver 300 of the space-time code approach, receiver 311 processessignals via antennas 309 and includes a demodulator 315, a decoder 317,and a memory 319. Unlike receiver 300, receiver 311 employs an OFDMfront-end 313, and includes a fast Fourier transform (FFT) logic 321that may operate in parallel with the demodulator 315.

The design of space-frequency codes for the OFDM based design approachis described below. These space-frequency codes optimally exploit bothspatial and frequency-selective diversity available in themulti-input-multi-output (MIMO) block fading channel. As in the singlecarrier time domain design approach, attention is focused on trellisbased codes because of the availability of reasonable complexity MLdecoders. For the purpose of explanation, the discussion pertains toBPSK modulated systems; however, it is recognized by one of ordinaryskill in the art that QPSK codes can be obtained by lifting the BPSKcodes, as described previously.

The general case in which C is a binary convolutional code of ratek/L_(t)L_(ISI).is considered. The encoder 203 processes k binary inputsequences x₁(t), x₂(t), . . . , x_(k)(t) and produces L_(t)L_(ISI) codedoutput sequences y₁(t), y₂(t), . . . ,y_(L) _(t) _(L) _(ISI) (t), whichare multiplexed together to form the output code word. The encoderaction is summarized by the following matrix equation

Y(D) = X(D)G(D), whereY(D) = ⌊Y₁(D)Y₂(D)  …  Y_(L_(t)L_(ISI))(D)⌋, X(D) = [X₁(D)X₂(D)  …  X_(k)(D)], and${G(D)} = \begin{bmatrix}{G_{1,1}(D)} & {G_{1,2}(D)} & \ldots & {G_{1,{L_{t}L_{ISI}}}(D)} \\{G_{2,1}(D)} & {G_{2,2}(D)} & \ldots & {G_{2,{L_{t}L_{ISI}}}(D)} \\\vdots & \vdots & ⋰ & \vdots \\{G_{k,1}(D)} & {G_{k,2}(D)} & \ldots & {G_{k,{L_{t}L_{ISI}}}(D)}\end{bmatrix}$

The natural space-time formatting of C is such that the output sequencecorresponding to Y_((m−1)L) _(t+1) (D)is assigned to the l^(th) transmitantenna in the m^(th) fading block. The algebraic analysis techniqueconsiders the rank of matrices formed by concatenating linearcombinations of the column vectors

${F_{l}(D)} = \begin{bmatrix}{G_{1,l}(D)} \\{G_{2,l}(D)} \\\vdots \\{G_{k,l}(D)}\end{bmatrix}$

is defined to be the set of binary full rank matrices

G

G=└g_(i,j)┘_(L) _(t) _(×L) _(t)

 resulting from applying any number of simple row operations to theidentity matrix I_(L) _(t) ; and ∀G₁ε

, 1≦i≦L_(t)1≦i≦L_(ISI),

${R_{i}^{({G_{m},m})}(D)} = {\lbrack {{{g_{i,1}(m)}I_{k}},{{g_{i,2}(m)}I_{k}},\ldots\mspace{11mu},{{g_{i,L_{t}}(m)}I_{k}}} \rbrack\begin{bmatrix}{F_{{{({m - 1})}L_{t}} + 1}(D)} \\{F_{{{({m - 1})}L_{t}} + 2}(D)} \\\vdots \\{F_{m\; L_{t}}(D)}\end{bmatrix}}$

Accordingly, the following algebraic construction for BPSKspace-frequency convolutional codes results. In a MIMO OFDM basedcommunication system with L_(t) transmit antennas 207 operating over afrequency selective block fading channel with L_(ISI) blocks, C denotesthe space-frequency code that includes the binary convolutional code C,whose k×L_(t)L_(ISI) transfer function matrix is G(D)=└F₁(D) . . . F_(L)_(t) _(L) _(ISI) (D)┘ and the spatial parser σ in which the outputY_((m−1)L) _(t) _(+l)(D)=X(D)F_(m−1)L) _(t) _(+l)(D) is assigned toantenna l in fading block m. Then, for BPSK transmission, C achieves dlevels of transmit diversity if d is the largest integer such that

  ∀G₁ ∈ ??, …  , G_(L_(ISI)) ∈ ??, 0 ≤ m₁ ≤ min (L_(t), L_(ISI)L_(t) − d + 1), …  , 0 ≤ m_(L_(ISI)) ≤ min (L_(t), L_(ISI)L_(t) − d + 1),  and$\mspace{20mu}{{{\sum\limits_{i = 1}^{L_{ISI}}m_{i}} = {{L_{ISI}L_{t}} - d + 1}},{{R_{m_{1},\mspace{11mu}\ldots\mspace{11mu},{m\; L_{ISI}}}^{({G_{1},\mspace{11mu}\ldots\mspace{11mu},G_{L_{ISI}}})}(D)} = \begin{bmatrix}{{R_{0}^{({G_{1},1})}(D)},\ldots\mspace{11mu},{R_{m_{1}}^{({G_{1},1})}(D)},} \\{{R_{0}^{({G_{2},2})}(D)},\ldots\mspace{11mu},{R_{m_{2}}^{({G_{2},2})}(D)},\ldots\mspace{11mu},{R_{m_{L_{ISI}}}^{({G_{L_{ISI}},L_{ISI}})}(D)}}\end{bmatrix}}}$has a rank k over the space of all formal series.

The above result allows for constructing convolutional space-frequencycodes that realize the optimum tradeoff between transmission rate anddiversity order for BPSK modulation with arbitrary coding rate, numberof transmit antenna, and number of fading blocks. It is readily seenthat this framework encompasses as a special case rate 1/n′convolutional codes with bit or symbol interleaving across the transmitantennas and frequency fading blocks.

Similar to the space-time coding approach, rate 1/L_(t) convolutionalcodes are considered, wherein the same transmission throughput isachieved. The output sequence from the ith arm Y_(i)(D) is assigned tothe ith antenna. The input assigned to each antenna 207 is thendistributed across the different fading blocks using a periodic bitinterleaver 209. The design of interleaver 209 depends largely onwhether the number of resolvable paths is available at the transmitter200. In the case in which this information is available at thetransmitter 200, the interleaver mapping function π is defined as

${{\pi(i)} = {\lbrack \frac{i}{L_{ISI}} \rbrack + {\frac{N}{L_{ISI}}(i)_{L_{ISI}}}}},$where ( )_(m) refers to the modulo m operation, 0≦i≦N−1, and N is thecode word length, which is assumed to be a multiple of L_(ISI).

In the absence of the prior information on the number of resolvablepaths in the channel 103, an interleaving scheme that is capable ofexploiting all the frequency diversity, whenever available, for anarbitrary unknown number of paths is needed. In the special case inwhich the number of paths is restricted to L_(ISI)=2^(r) (for anyarbitrary integer r) and the maximum possible number of paths L_(ISI)^((max)) is known at the transmitter 200, the following construction forthe universal interleaving map is provided:

${{\pi(i)} = {{\sum\limits_{k = 0}^{\log_{2}{(L_{ISI}^{(\max)})}}{a_{k}\frac{N}{2^{k + 1}}}} + \lbrack \frac{i}{L_{ISI}^{(\max)}} \rbrack}},{a_{k} = ( \frac{{(i)L_{ISI}^{(\max)}} - {\sum\limits_{j = 0}^{k - 1}{a_{j}2^{j}}}}{2^{k}} )}$This interleaving scheme distributes the input sequence periodicallyamong the L_(ISI) fading blocks for any L_(ISI)=2^(r) andL_(ISI)≦L_(ISI) ^((max)). In practical applications, L_(ISI) ^((max))may be chosen to be larger than the maximum number of resolvable pathsexpected in this particular application, and hence, the transmitter 200does not need feedback from the receiver 300. This does not result inany loss of performance. If the number of paths is not a power of two,then the diversity advantage is lower bounded by that achieved with thenumber of paths equal to L_(ISI) ^((approx))) such that L_(ISI)^((approx))=2^(r)<L_(ISI).

Table 3 shows the diversity advantage that is achieved by the optimalfree distance codes when used as space-frequency codes in this scenario.Specifically, Table 3 lists the diversity advantage for BPSK algebraicspace-frequency codes with optimal free distance for MIMO frequencyselective fading channels.

TABLE 3 d for d for d for d for L_(t) ν Connection Polynomials L_(ISI) =1 L_(ISI) = 2 L_(ISI) = 3 L_(ISI) = 4 2 2 5, 7 2 4 5 6 3 64, 74 2 4 6 74 46, 72 2 4 6 8 5 65, 57 2 4 6 8 6 554, 744 2 4 6 8 3 3 54, 64, 74 3 4— — 4 52, 66, 76 3 3 5 — 5 47, 53, 75 3 — — — 6 554, 624, 764 3 — — — 44 52, 56, 66, 76 4 — — — 5 53, 67, 71, 75 4 — — — 5 5 75, 71, 73, 65, 575 — — —While, the codes in Table 3 may not realize the maximum possiblediversity advantage under all circumstances, these codes a compromisebetween the diversity advantage and coding gain.

The OFDM based approach addresses the need for a lower complexitymaximum likelihood receiver 300. This approach recognizes the fact thatthe maximum likelihood decoder 317 complexity in the OFDM approach doesnot increase exponentially with the number of resolvable paths, contraryto the space-time coding approach. It should be noted that this does notmean, however, that complexity of the decoder 317 does not depend on thenumber of paths. As shown in Table 3, as the number of paths increases,the codes with larger constraint lengths are needed to efficientlyexploit the diversity available in the channel 103. Unlike thespace-time coding approach, it is possible to trade diversity advantagefor a reduction in complexity by choosing a code with a small constraintlength. This trade-off is not possible in the space-time coding approachbecause, irrespective of the constraint length of the code, thecomplexity of the (ML) decoder 305 grows exponentially with the numberof resolvable paths. The OFDM based approach, however, provides arelatively lower diversity advantage over the space-time codingapproach.

The maximum transmit diversity advantage achieved in a BPSK OFDM MIMOwireless system with L_(t) transmit antennas 207 and L_(ISI) resolvablepaths/antenna supporting a throughput of 1 bps/Hz is L_(ISI)(L_(t)−1)+1.It is clear that the maximum diversity advantage under this approach islower as compared to the space-time coding approach (i.e, L_(t)L_(ISI)).The results in Tables 1 and 3 compare the diversity advantage achievedby space-time codes and space-frequency codes for different values ofL_(t) and L_(ISI). As will be evident from the discussion below, thisloss in diversity advantage may not always lead to a performance loss inthe frame error rate range of interest.

FIGS. 4A-4H show graphs of simulation results of the channel codes, inaccordance with the various embodiments of the present invention.Specifically, these figures show the simulated frame error rateperformance results for the two coding approaches, concentrating on thecodes presented in Tables 1, 2, and 3. In all cases, the frame lengthcorresponds to 100 simultaneous transmissions from all antennas 207.Joint maximum likelihood decoding and equalization that accounts for theISI nature of the channel is assumed at the receiver (e.g., 300 and311). In most cases, the simulated frame error rates were restricted toless than 1% because of the practical significance of this range and tolimit the simulation time.

FIGS. 4A-4G report the performance of the two proposed approaches inBPSK systems with different numbers of transmit antennas L_(t), receiveantennas L_(r), resolvable paths L_(ISI), and receiver trelliscomplexity. The number of states in the figures represents the maximumlikelihood decoder trellis complexity. For the OFDM approach, thisnumber is equal to the number of states in the underlying convolutionalcodes; however, for the space-time coding approach, this number accountsfor the additional complexity dictated by the ISI nature of the channel.In the figures, the single carrier approach with space-time coding isreferred to as (STC), whereas the OFDM approach with space-frequencycoding is referred to as (SFC).

In FIGS. 4A and 4B, the gain in performance of the two approaches areshown with respect to an increasing number of resolvable paths. In thesingle carrier approach, this improvement provides a concomitantincrease in receiver complexity as the number of states in the maximumlikelihood receiver grows exponentially with the number of resolvablepaths. In contrast, for the space-frequency coding approach, theperformance improvement does not entail any increase in complexity. Itis noted that the improvement in performance in the SFC approach ismarginal when L_(ISI) increases from one to two because, as shown inTable 1; the diversity advantage of the 4-state code used is the same inboth scenario.

FIGS. 4C-4F provides a comparison between the STC and SFC approaches. Itis shown that when the same code is used in both schemes, the STCapproach always provides a gain in performance, however, at the expenseof higher receiver complexity. Whereas, if the receiver complexity isfixed in both approaches, the SFC approach sometimes offers betterperformance. This may seem in contrary to the intuition based on thesuperiority of the STC approach in terms of diversity advantage; thisseeming contradiction can be attributed to two reasons. First, the samereceiver complexity allows the SFC approach to utilize moresophisticated codes that offer larger coding gains. Second, the effectof the STC superior diversity advantage may only become apparent atsignificantly larger signal-to-noise ratios. This observation, however,indicates that the SFC approach may yield superior performance in somepractical applications.

FIG. 4G highlights the importance of careful design in optimizing thediversity advantage. In this figure, the 4-state (5,7) optimal freedistance SFC is compared with the 4-state (6,7) in a system withL_(t)=−2, L_(r)=I, and L_(ISI)=2,3. As reported in Table 1, the (5,7)code achieves d=2,3 for L_(ISI)=2,3, respectively. Whereas, the (6,7)code achieves d=3 in both codes; it is noted that in the L_(ISI), d=3 isthe maximum possible diversity advantage for this throughput. As shownin the figure, for the L_(ISI)=2 case, the superior diversity advantageof the (6,7) is apparent in the steeper frame error rate curve slope.This results in a gain of about 1 dB at 0.01 frame error rate. On theother hand, for the L_(ISI)=3 case, it is shown that the (5,7) codeexhibits a superior product distance that accounts for about 1 dB gaincompared with the (6,7) code.

The above construct has applicability in a number of communicationsystems; for example, the developed channel codes can be deployed in awireless communication, as seen in FIG. 5.

FIG. 5 shows a diagram of a wireless communication system that utilizesthe channel codes, according to the various embodiments of the presentinvention. In a wireless communication system 500, multiple terminals501 and 503 communicate over a wireless network 505. Terminal 501 isequipped with an encoder 203 (as shown in FIG. 2) that generatesspace-time or space-frequency codes. Terminal 501 also includes multipletransmit antennas 207 (as shown in FIG. 2). In this example, each of theterminals 501 and 503 are configured to encode and decode the space-timecodes; accordingly, both of the terminals 501 and 503 possess thetransmitter 200 and receiver 300. However, it is recognized that each ofthe terminals 501 and 503 may alternatively be configured as atransmitting unit or a receiving unit, depending on the application. Forexample, in a broadcast application, terminal 501 may be used as ahead-end to transmit signals to multiple receiving terminals (in whichonly receiving terminal 503 is shown). Consequently, terminal 503 wouldonly be equipped with a receiver 300. Alternatively, each of theterminals 501 and 503 may be configured to operate using space-frequencycodes. As mentioned previously, the choice of space-time codes versusspace-frequency codes depends largely on the trade-off between receivercomplexity and the desired diversity advantage.

FIG. 6 shows a diagram of a computer system that can perform theprocesses of encoding and decoding of the channel codes, in accordancewith the embodiments of the present invention. Computer system 601includes a bus 603 or other communication mechanism for communicatinginformation, and a processor 605 coupled with bus 603 for processing theinformation. Computer system 601 also includes a main memory 607, suchas a random access memory (RAM) or other dynamic storage device, coupledto bus 603 for storing information and instructions to be executed byprocessor 605. In addition, main memory 607 may be used for storingtemporary variables or other intermediate information during executionof instructions to be executed by processor 605. Computer system 601further includes a read only memory (ROM) 609 or other static storagedevice coupled to bus 603 for storing static information andinstructions for processor 605. A storage device 611, such as a magneticdisk or optical disk, is provided and coupled to bus 603 for storinginformation and instructions.

Computer system 601 may be coupled via bus 603 to a display 613, such asa cathode ray tube (CRT), for displaying information to a computer user.An input device 615, including alphanumeric and other keys, is coupledto bus 603 for communicating information and command selections toprocessor 605. Another type of user input device is cursor control 617,such as a mouse, a trackball, or cursor direction keys for communicatingdirection information and command selections to processor 605 and forcontrolling cursor movement on display 613.

According to one embodiment, channel code generation within system 100is provided by computer system 601 in response to processor 605executing one or more sequences of one or more instructions contained inmain memory 607. Such instructions may be read into main memory 607 fromanother computer-readable medium, such as storage device 611. Executionof the sequences of instructions contained in main memory 607 causesprocessor 605 to perform the process steps described herein. One or moreprocessors in a multi-processing arrangement may also be employed toexecute the sequences of instructions contained in main memory 607. Inalternative embodiments, hard-wired circuitry may be used in place of orin combination with software instructions. Thus, embodiments are notlimited to any specific combination of hardware circuitry and software.

Further, the instructions to support the generation of space-time codesand space-frequency codes of system 100 may reside on acomputer-readable medium. The term “computer-readable medium” as usedherein refers to any medium that participates in providing instructionsto processor 605 for execution. Such a medium may take many forms,including but not limited to, non-volatile media, volatile media, andtransmission media. Non-volatile media includes, for example, optical ormagnetic disks, such as storage device 611. Volatile media includesdynamic memory, such as main memory 607. Transmission media includescoaxial cables, copper wire and fiber optics, including the wires thatcomprise bus 603. Transmission media can also take the form of acousticor light waves, such as those generated during radio wave and infrareddata communication.

Common forms of computer-readable media include, for example, a floppydisk, a flexible disk, hard disk, magnetic tape, or any other magneticmedium, a CD-ROM, any other optical medium, punch cards, paper tape, anyother physical medium with patterns of holes, a RAM, a PROM, and EPROM,a FLASH-EPROM, any other memory chip or cartridge, a carrier wave asdescribed hereinafter, or any other medium from which a computer canread.

Various forms of computer readable media may be involved in carrying oneor more sequences of one or more instructions to processor 605 forexecution. For example, the instructions may initially be carried on amagnetic disk of a remote computer. The remote computer can load theinstructions relating to encoding and decoding of space-time codes usedin system 100 remotely into its dynamic memory and send the instructionsover a telephone line using a modem. A modem local to computer system601 can receive the data on the telephone line and use an infraredtransmitter to convert the data to an infrared signal. An infrareddetector coupled to bus 603 can receive the data carried in the infraredsignal and place the data on bus 603. Bus 603 carries the data to mainmemory 607, from which processor 605 retrieves and executes theinstructions. The instructions received by main memory 607 mayoptionally be stored on storage device 611 either before or afterexecution by processor 605.

Computer system 601 also includes a communication interface 619 coupledto bus 603. Communication interface 619 provides a two-way datacommunication coupling to a network link 621 that is connected to alocal network 623. For example, communication interface 619 may be anetwork interface card to attach to any packet switched local areanetwork (LAN). As another example, communication interface 619 may be anasymmetrical digital subscriber line (ADSL) card, an integrated servicesdigital network (ISDN) card or a modem to provide a data communicationconnection to a corresponding type of telephone line. Wireless links mayalso be implemented. In any such implementation, communication interface619 sends and receives electrical, electromagnetic or optical signalsthat carry digital data streams representing various types ofinformation.

Network link 621 typically provides data communication through one ormore networks to other data devices. For example, network link 621 mayprovide a connection through local network 623 to a host computer 625 orto data equipment operated by a service provider, which provides datacommunication services through a communication network 627 (e.g., theInternet). LAN 623 and network 627 both use electrical, electromagneticor optical signals that carry digital data streams. The signals throughthe various networks and the signals on network link 621 and throughcommunication interface 619, which carry the digital data to and fromcomputer system 601, are exemplary forms of carrier waves transportingthe information. Computer system 601 can transmit notifications andreceive data, including program code, through the network(s), networklink 621 and communication interface 619.

The techniques described herein provide several advantages over priorapproaches to providing space-time codes. The two approaches ofdesigning space-time codes and space-frequency codes optimally exploitsboth the spatial and frequency diversity available in the channel.

Obviously, numerous modifications and variations of the presentinvention are possible in light of the above teachings. It is thereforeto be understood that within the scope of the appended claims, theinvention may be practiced otherwise than as specifically describedherein.

REFERENCES

[1] E. Teletar. Capacity of Multi-Antenna Gaussian Channels. TechnicalReport, AT&T-Bell Labs, June 1995.

[2] G. J. Foschini and M. Gans. On the Limits of Wireless Communicationin a Fading Environment When Using Multiple Antennas. Wireless PersonalCommunication, 6:311-335, March 1998.

[3] V. Tarokh, N. Seshadri, and A. R. Calderbank. Space-Time Codes forHigh Data Wireless Communication: Performance Criterion and CodeConstruction. IEEE Trans. Info. Theory, IT-44:774-765, March 1998.

[4] J.-C. Guey, M. R. Bell M. P. Fitz, and W.-Y. Kuo. Signal Design forTransmitter Diversity, Wireless Communication Systems over RayleighFading Channels. IEEE Vehicular Technology Conference, pages 136-140,Atlanta, 1996.

[5] G. J. Foschini. Layered Space-Time Architecture for WirelessCommunication in Fading Environments When Using Multiple Antennas. BellLabs Tech. J., 2, Autumn 1996.

[6] S. Lin and Jr. D. J. Costello. Error Control Coding: Fundamentalsand Applications. Prentice-Hall, New Jersey, 1983.

1. A method comprising: demodulating a signal containing a code word,the code word having a construction that defines a plurality of pathsassociated with an intersymbol interference (ISI) environment of acommunication channel, the code word achieving a diversity based uponthe number of transmit antennas and the number of ISI paths, wherein thecode word is decoded to output a message signal.
 2. The method accordingto claim 1, wherein the code word satisfies a baseband rank criterionsuch that rank(f_(ISI)(c)−f_(ISI)(e)) is maximized over all pairs ofdistinct code words c, e∈C, C being an L_(t)×l linear space-time code,L_(t) representing the number of transmit antennas, wherein${{f(c)}_{ISI} = \begin{bmatrix}{f(c)} & \underset{\_}{0} & \ldots & \underset{\_}{0} \\\underset{\_}{0} & {f(c)} & \ldots & \underset{\_}{0} \\\vdots & \vdots & ⋰ & \vdots \\\underset{\_}{0} & \underset{\_}{0} & \ldots & {f(c)}\end{bmatrix}},$ 0 being an L_(t)×1 all zero vector.
 3. The methodaccording to claim 2, wherein rank(f_(ISI)(c)−f_(ISI)(e))=L_(t)L_(ISI)for all pairs of distinct code words c, e∈C, and L_(ISI) represents thenumber of ISI paths.
 4. The method according to claim 2, wherein theconstruction further defines an ISI code word matrix as follows:${c_{ISI} = \begin{bmatrix}c & \underset{\_}{0} & \ldots & 0 \\0 & c & \ldots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \ldots & c\end{bmatrix}},$ wherein f(c_(ISI))≠f(c)_(ISI), the diversity beingbased upon f(c_(ISI))−f(e_(ISI))=f(c)_(ISI)−f(e)_(ISI).
 5. The methodaccording to claim 4, wherein the construction further defines an ISIchannel binary rank criterion such that C has an underlying binary codeC of length N=L_(t)l operating in the ISI environment andl≧L_(t)L_(ISI), L_(ISI) representing the number of ISI paths.
 6. Themethod according to claim 5, wherein, for every non-zero code word ccorresponding to C_(ISI) of full rank L_(t)L_(ISI) over a binary fieldF, the diversity of the space-time code C is L_(t)L_(ISI).
 7. The methodaccording to claim 1, wherein the construction further defines M₁, M₂, .. . , M_(L) _(t) as binary matrices of dimension k×l,l≧k, and C is anL_(t)×l linear space-time code of dimension k and includes code wordmatrices defined as follows: ${c = \begin{bmatrix}{\underset{\_}{x}M_{1}} \\{\underset{\_}{x}M_{2}} \\\vdots \\{\underset{\_}{x}M_{L_{t}}}\end{bmatrix}},$ wherein x denotes an arbitrary k-tuple of informationbits associated with the message signal, and L_(t)<l, L_(t) representingthe number of transmit antennas.
 8. The method according to claim 7,wherein the construction further defines M_(n,m)=└O_(L) _(t)_(×(m−1))M_(n)O_(L) _(t) _(×(L) _(ISI) _(+1−m))┘, O_(L) _(t)_(×(m−)being an L_(t)×(m−1) all zero matrix, for BPSK transmission, thediversity is L_(t)L_(ISI), if and only if M_(1,1), M_(2,1), . . . ,M_(L) _(t) _(L) _(ISI) , ∀α₁,α₂, . . . ,α_(L) _(t) ∈F: M=α₁M_(1,1)⊕α₂M_(2,1) ⊕. . . ⊕a_(L) _(t) _(L) _(ISI) M_(L) _(t) _(L) _(ISI) is offull rank k unless α₁=. . . ,α_(L) _(t) _(L) _(ISI) =0, F being a binaryfield, the code word being drawn from $c_{ISI} = {\begin{bmatrix}{\underset{\_}{x}M_{1,1}} \\{\underset{\_}{x}M_{1,2}} \\\vdots \\{\underset{\_}{x}M_{L_{t},L_{ISI}}}\end{bmatrix}.}$
 9. The method according to claim 1, wherein thereceived signal is modulated using at least one of BPSK (binaryphase-shift keying) modulation and QPSK (quadrature phase-shift keying)modulation.
 10. The method according to claim 1, wherein the decoderutilizes a maximum likelihood decoding algorithm to decode the receivedsignal.
 11. The method according to claim 1, further comprising: storingchannel state information of the communication channel, wherein the codeword is decoded based upon the channel state information.
 12. A devicecomprising: demodulating a signal containing a code word, the code wordhaving a construction that defines a plurality of paths associated withan intersymbol interference (ISI) environment of a communicationchannel, the code word achieving a diversity based upon the number oftransmit antennas and the number of ISI paths, wherein the code word isdecoded to output a message signal.
 13. The device according to claim12, wherein the code word satisfies a baseband rank criterion such thatrank(f_(ISI)(c)f_(ISI)(e)) is maximized over all pairs of distinct codewords c, e∈C, C being an L₁× linear space-time code, L_(t), representingthe number of transmit antennas, wherein${{f(c)}_{ISI} = \begin{bmatrix}{f(c)} & \underset{\_}{0} & \ldots & \underset{\_}{0} \\\underset{\_}{0} & {f(c)} & \ldots & \underset{\_}{0} \\\vdots & \vdots & ⋰ & \vdots \\\underset{\_}{0} & \underset{\_}{0} & \ldots & {f(c)}\end{bmatrix}},$ 0 being an L_(t)×1 all zero vector.
 14. The deviceaccording to claim 13, wherein rank(f_(ISI)(c)−f_(ISI)(e))=L_(t)L_(ISI)for all pairs of distinct code words c, e∈C, and L_(ISI) represents thenumber of ISI paths.
 15. The device according to claim 13, wherein theconstruction further defines an ISI code word matrix as follows:${c_{ISI} = \begin{bmatrix}c & \underset{\_}{0} & \ldots & 0 \\0 & c & \ldots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \ldots & c\end{bmatrix}},$ wherein f(c_(ISI))≠f(c)_(ISI), the diversity beingbased upon f(c_(ISI))−f(e_(ISI))=f(c)_(ISI)−f(e)_(ISI).
 16. The deviceaccording to claim 15, wherein the construction further defines an ISIchannel binary rank criterion such that C has an underlying binary codeC of length N=L_(t)l operating in the ISI environment andl≧L_(t)L_(ISI), L_(ISI) representing the number of ISI paths.
 17. Thedevice according to claim 16, wherein, for every non-zero code word ccorresponding to c_(ISI) of full rank L_(t)L_(ISI) over a binary fieldF, the diversity of the space-time code C is L_(t)L_(ISI).
 18. Thedevice according to claim 12, wherein the construction further definesM₁, M₂, . . . , M_(L) _(t) as biny matrices of dimension k×l,l≧k, and Cis an L_(t)×l linear space-time code of dimension k and includes codeword matrices defined as follows: ${c = \begin{bmatrix}{\underset{\_}{x}M_{1}} \\{\underset{\_}{x}M_{2}} \\\vdots \\{\underset{\_}{x}M_{L_{t}}}\end{bmatrix}},$ wherein x denotes an arbitrary k-tuple of informationbits associated with the message signal, and L_(t)<l, L_(t),representing the number of transmit antennas.
 19. The device accordingto claim 18, wherein the construction further defines M_(n,m)=└O_(L)_(t) _(×(m−1))M_(n)O_(L) _(t) _(×(L) _(ISI) _(=l−m)┘, O) _(L) _(t)_(×(m−l))being an L_(t)×(m −1) all zero matrix, for BPSK transmission,the diversity is L_(t)L_(ISI), if and only if M_(1,l),M _(2,1), . . .,M_(L) _(t) _(L) _(ISI) , ∀α₁,α₂, . . . ,α_(L) _(t) ∈ F:M=α_(l)M_(l,l)⊕α₂M_(2,l)⊕. . . ⊕α_(L) _(t) _(L) _(ISI) is of full rank kunless α₁=. . . α_(L) _(t) _(L) _(ISI) =0, F being a binary field, thecode word being drawn from $c_{ISI} = {\begin{bmatrix}{\underset{\_}{x}M_{1,1}} \\{\underset{\_}{x}M_{1,2}} \\\vdots \\{\underset{\_}{x}M_{L_{t},L_{ISI}}}\end{bmatrix}.}$
 20. The device according to claim 12, wherein thereceived signal is modulated using at least one of BPSK (binaryphase-shift keying) modulation and QPSK (quadrature phase-shift keying)modulation.
 21. The device according to claim 12, wherein the decoderutilizes a maximum likelihood decoding algorithm to decode the receivedsignal.
 22. The device according to claim 12, wherein channel stateinformation of the communication channel is stored and used to decodethe code word.
 23. A method comprising: receiving a code word having aconstruction that defines a plurality of paths associated with anintersymbol interference (IDI) environment of a communication channel,the code word achieving a diversity based upon the number of transmitantennas and the number of ISI paths, wherein the code word is decodedto output a message signal.
 24. A system comprising: means for receivinga code word having a construction that defines a plurality of pathsassociated with an intersymbol interference (ISI) environment of acommunication channel, the code word achieving a diversity based uponthe number of transmit antennas and the number of ISI paths, wherein thecode word is decoded to output a message signal.